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Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1967 Swedish Mathematical Competition
4
4
Part of
1967 Swedish Mathematical Competition
Problems
(1)
lim b_n = 0 and sum a_ib_i$ diverges when sum a_i diverges
Source: 1967 Swedish Mathematical Competition p4
3/21/2021
The sequence
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, ...
a
1
,
a
2
,
a
3
,
...
of positive reals is such that
∑
a
i
\sum a_i
∑
a
i
diverges. Show that there is a sequence
b
1
,
b
2
,
b
3
,
.
.
.
b_1, b_2, b_3, ...
b
1
,
b
2
,
b
3
,
...
of positive reals such that
lim
b
n
=
0
\lim b_n = 0
lim
b
n
=
0
and
∑
a
i
b
i
\sum a_ib_i
∑
a
i
b
i
diverges.
algebra
Sum
limit